2.2 Critical Behavior in Collapse2 Singularities in AF Spacetimes2 Singularities in AF Spacetimes

2.1 Naked Singularities and the Hoop Conjecture

2.1.1 Overview

Perhaps the first numerical approach to study the cosmic censorship conjecture consisted of attempts to create naked singularities. Many of these studies were motivated by Thorne's ``hoop conjecture'' [175Jump To The Next Citation Point In The Article] that collapse will yield a black hole only if a mass M is compressed to a region with circumference tex2html_wrap_inline1258 in all directions. (As is discussed by Wald [178Jump To The Next Citation Point In The Article], one runs into difficulties in any attempt to formulate the conjecture precisely. For example, how does one define C and M, especially if the initial data are not at least axially symmetric?) If the hoop conjecture is true, naked singularities may form if collapse can yield tex2html_wrap_inline1264 in some direction. The existence of a naked singularity is inferred from the absence of an apparent horizon (AH) which can be identified locally by following null geodesics. Although a definitive identification of a naked singularity requires the event horizon (EH) to be proven to be absent, to identify an EH requires knowledge of the entire spacetime. While one finds an AH within an EH [120, 121], it is possible to construct a spacetime slicing which avoids the AH even though an EH is present [180Jump To The Next Citation Point In The Article]. Methods to find an EH in a numerically determined spacetime have only recently become available and have not been applied to this issue [132Jump To The Next Citation Point In The Article, 136Jump To The Next Citation Point In The Article].

2.1.2 Naked Spindle Singularities?

In the best known attempt to produce naked singularities, Shapiro and Teukolsky (ST) [169Jump To The Next Citation Point In The Article] considered collapse of prolate spheroids of collisionless gas. (Nakamura and Sato [146] had previously studied the collapse of non-rotating deformed stars with an initial large reduction of internal energy and apparently found spindle or pancake singularities in extreme cases.) ST solved the general relativistic Vlasov equation for the particles along with Einstein's equations for the gravitational field. Null geodesics were followed to identify an AH if present. The curvature invariant tex2html_wrap_inline1266 was also computed. They found that an AH (and presumably a BH) formed if tex2html_wrap_inline1268 everywhere but no AH (and presumably a naked singularity) in the opposite case. In the latter case, the evolution (not surprisingly) could not proceed past the moment of formation of the singularity. In a subsequent study, ST [170] also showed that a small amount of rotation (counter rotating particles with no net angular momentum) does not prevent the formation of a naked spindle singularity. However, Wald and Iyer [180] have shown that the Schwarzschild solution has a time slicing whose evolution approaches arbitrarily close to the singularity with no AH in any slice, (but of course, with an EH in the spacetime). This may mean that there is a chance that the increasing prolateness found by ST in effect changes the slicing to one with no apparent horizon just at the point required by the hoop conjecture. While, on the face of it, this seems unlikely, Tod gives an example where the AH does not form on a chosen constant time slice--but rather different portions form at different times. He argues that a numerical simulation might be forced by the singularity to end before the formation of the AH is complete. Such an AH would not be found by the simulations [177Jump To The Next Citation Point In The Article]. In response to such a possibility, Shapiro and Teukolsky considered equilibrium sequences of prolate relativistic star clusters [171]. The idea is to counter the possibility that an EH might form after the simulation must stop. If an equilibrium configuration is non-singular, it cannot contain an EH, since singularity theorems say that an EH implies a singularity. However, a sequence of non-singular equilibria with rising I ever closer to the spindle singularity would lend support to the existence of a naked spindle singularity since one can approach the singular state without formation of an EH. They constructed this sequence and found that the singular end points were very similar to their dynamical spindle singularity. Wald believes, however, that it is likely that the ST slicing is such that their singularities are not naked--the AH is present but has not yet appeared in their time slices [178].

Another numerical study of the hoop conjecture was made by Chiba et al [58]. Rather than a dynamical collapse model, they searched for AH's in analytic initial data for discs, annuli, and rings. Previous studies of this type were done by Nakamura et al [147] with oblate and prolate spheroids and by Wojtkiewicz [182] with axisymmetric singular lines and rings. The summary of their results is that an AH forms if tex2html_wrap_inline1272 . (Analytic results due to Barrabès et al [5, 4] and Tod [177] give similar quantitative results with different initial data classes and (possibly) definition of C .)

There is strong analytical evidence against the development of spindle singularities. It has been shown by Chrusciel and Moncrief that strong cosmic censorship holds in AF electrovac solutions which admit a tex2html_wrap_inline1276 symmetric Cauchy surface [24Jump To The Next Citation Point In The Article]. The evolutions of these highly nonlinear equations are in fact non-singular.

2.1.3 Going Further

Motivated by ST's results [169], Echeverria [70] numerically studied the properties of the naked singularity that is known to form in the collapse of an infinite, cylindrical dust shell [175]. While the asymptotic state can be found analytically, the approach to it must be followed numerically. The analytic asymptotic solution can be matched to the numerical one (which cannot be followed all the way to the collapse) to show that the singularity is strong (an observer experiences infinite stretching parallel to the symmetry axis and squeezing perpendicular to the symmetry axis). A burst of gravitational radiation emitted just prior to the formation of the singularity stretches and squeezes in opposite directions to the singularity. This result for dust conflicts with rigorously nonsingular solutions for the electrovac case [24]. One wonders then if dust collapse gives any information about singularities of the gravitational field.

Nakamura et al (NSN) [148] conjectured that even if naked spindle singularities could exist, they would either disappear or become black holes. This demise of the naked singularity would be caused by the back reaction of the gravitational waves emitted by it. While NSN proposed a numerical test of their conjecture, they believed it to be beyond the current generation of computer technology.

The results of all these searches for naked spindle singularities are controversial, but they could be resolved if the presence or absence of the EH could be determined. One could demonstrate numerically whether or not Wald's view of ST's results is correct by using existing EH finders [132, 136] in a relevant simulation.


2.2 Critical Behavior in Collapse2 Singularities in AF Spacetimes2 Singularities in AF Spacetimes

image Numerical Approaches to Spacetime Singularities
Beverly K. Berger
http://www.livingreviews.org/lrr-1998-7
© Max-Planck-Gesellschaft. ISSN 1433-8351
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